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Use the dataset BodyFat, which gives the percent of weight made up of body fat for 100 men as well as other variables such as Age, Weight (in pounds), Height (in inches), and circumference (in \(\mathrm{cm}\) ) measurements for the Neck, Chest, Abdomen, Ankle, Biceps, and Wrist. \(^{78}\) The regression line for predicting body fat percent using neck circumference is BodyFat \(=-47.9+1.75 \cdot\) Neck (a) What body fat percent does the line predict for a person with a neck circumference of \(35 \mathrm{~cm}\) ? Of \(40 \mathrm{~cm} ?\) (b) Interpret the slope of the line in context. (c) One of the men in the study had a neck circumference of \(38.7 \mathrm{~cm}\) and a body fat percent of \(11.3 .\) Find the residual for this man.

Step by step solution

01

Predict Body Fat% Given Neck Circumference

To predict the body fat percent for a person with a neck circumference of 35 cm and 40 cm, substitute these values into the regression equation. So, for a neck circumference of 35 cm, Body Fat %= \(-47.9+(1.75*35)\) and for a neck circumference of 40 cm, Body Fat% = \(-47.9+(1.75*40)\)

02

Interpret the slope

The slope of the line represents the change in the dependent variable (Body Fat%) for each unit increase in the independent variable (Neck Circumference). In this case, the slope is 1.75, which means for each cm increase in neck circumference, the body fat percent is predicted to increase by 1.75%.

03

Calculate the residual

To find the residual for the man with a neck circumference of 38.7 cm and a body fat percent of 11.3, first predict the body fat percent using the regression line as Body Fat% = \(-47.9 + (1.75 * 38.7)\). Now, subtract this predicted value from the actual value of body fat percent. The residual = Observed body fat% - Predicted body fat%

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Residuals

When performing regression analysis, residuals help us understand how well our model fits the data. A residual is the difference between the actual value and the predicted value from our regression equation. It shows how much the prediction is off.
Imagine a man in the study with a neck circumference of 38.7 cm and a body fat percent of 11.3%. To determine how accurate our prediction is, we calculate the residual.

• First, use the regression line: \[ \text{Body Fat extpercent} = -47.9 + (1.75 \times 38.7) \]
• Calculate the predicted body fat percent and subtract it from the actual body fat percent (11.3%).
• The resulting value is the residual, indicating whether the prediction was too high or too low.

A small residual means the prediction is close to the actual value, indicating a good model fit.

Slope Interpretation

The slope in a regression line is a crucial number. It tells us how much the dependent variable changes for each unit change in the independent variable. In our example, the slope is 1.75.
But what does this mean in the context of predicting body fat percent?

• The "1.75" represents how much body fat percent is expected to change for every 1 cm increase in neck circumference.
• So, if neck circumference increases by 1 cm, the body fat percent is predicted to increase by 1.75%.

This interpretation helps in understanding the relationship between body fat and neck circumference. It shows that neck circumference is positively associated with body fat percent, meaning as neck size goes up, so does body fat percent.

Predictive Modeling

Predictive modeling, especially using regression analysis, is a crucial tool in statistics. It assists in making data-driven predictions about unknown outcomes. By looking at relationships between variables, models can predict future or unknown values.
In this exercise, we're using neck circumference to predict body fat percent.

• The regression equation: \[ \text{Body Fat extpercent} = -47.9 + 1.75 \times \text{Neck Circumference} \]
• This equation helps predict body fat based on a neck measurement. It becomes a helpful tool in medical and health-related fields.
• To predict, simply plug in the neck circumference value into the equation.

Predictive models like this are valuable because they reduce uncertainty and allow for informed decision-making based on statistical analysis.